3 Sept Irish Math Society

Transcription

1 3 Sept Irish Math Society Calculating the Weather: The Mathematics of Atmospheric Modelling Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin

2 Outline of the Lecture Pre-history of NWP Richardson s Forecast The ENIAC Integrations Interlude Data Assimilation Ensemble Prediction Spherical Grids 2

3 Increase in Forecasting Skill 3

4 Relevant Mathematical Areas Partial Differential Equations Numerical Analysis Linear Algebra Variational Methods Dynamical Systems Geometry of the Sphere Something for everyone! 4

5 Ancient Times 5

6 Galileo Galilei ( ) Galileo formulated the basic law of falling bodies, which he verified by careful measurements. He constructed a telescope, with which he studied lunar craters, and discovered four moons revolving around Jupiter. Galileo is credited with the invention of the Thermometer. Thus began quantitative measurements of the atmosphere. 6

7 Evangelista Torricelli Evangelista Torricelli ( ), a student of Galileo, devised the first accurate barometer. Torricelli s Theorem: v = 2gh Torricelli inventing the barometer 7

8 Newton s Law of Motion The rate of change of momentum of a body is equal to the sum of the forces acting on the body. If F is the total applied force, Newton s Second Law gives a differential equation: dp dt = F. The acceleration a is the rate of change of velocity, that is, a = dv/dt. If the mass m is constant, we have F = ma. Force = Mass Acceleration. 8

9 Euler s Equations for Fluid Flow Leonhard Euler Born in Basel in Died 1783 in St Petersburg. Formulated the equations for incompressible, inviscid fluid flow: V t + V V + 1 ρ p = g. V = 0 Partial differential equations. 9

10 Jean Le Rond d Alembert A body moving at constant speed through a gas or a fluid does not experience any resistance (D Alembert 1752). 10

11 George G Stokes, George Gabriel Stokes, founder of modern hydrodynamics. 11

12 ASIDE: Stokes Theorem Γ V dl = Σ V n da. Stokes Theorem was actually discovered by Kelvin in It is of central importance in fluid dynamics. It leads on to Bjerknes Circulation Theorem: dc dt = 1 ρ p da = dp ρ, Σ which generalized Kelvin s Circulation Theorem to baroclinic fluids (ρ varying independently of p), and ushered in the study of Geophysical Fluid Dynamics. Γ 12

13 Resolution of d Alembert s Paradox The minutest amount of viscosity has a profound qualitative impact on the character of the solution. The Navier-Stokes equations incorporate the effect of viscosity. 13

14 The Navier-Stokes Equations Euler s Equations: V t + V V + 1 ρ p = g. The Navier-Stokes Equations V t + V V + 1 ρ p = ν 2 V + g. Motion on the rotating Earth: V t + V V + 2Ω V + 1 ρ p = ν 2 V + g. 14

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16 The Inventors of Thermodynamics 16

17 The Primitive Equations ( du dt f + u tan φ ) v + 1 p a ρ x + F x = 0 ( dv dt + f + u tan φ ) u + 1 p a ρ y + F y = 0 p = RρT p z + gρ = 0 dt dt + (γ 1)T V = Q c p ρ t + ρv = 0 ρ w t + ρ wv = [Sources Sinks] Seven equations; seven variables (u, v, w, p, T, ρ, ρ w ). 17

18 Scientific Weather Forecasting in a Nut-Shell The atmosphere is a physical system Its behaviour is governed by the laws of physics These laws are expressed quantitatively in the form of mathematical equations Using observations, we can specify the atmospheric state at a given initial time: Today s Weather Using the equations, we can calculate how this state will change over time: Tomorrow s Weather The equations are very complicated (non-linear) and a powerful computer is required to do the calculations The accuracy decreases as the range increases; there is an inherent limit of predictibility. 18

19 Richardson s Forecast 19

20 Lewis Fry Richardson, During WWI, Richardson computed by hand the pressure change at a single point. It took him two years! His forecast was a catastrophic failure: p = 145 hpa in 6 hrs But Richardson s method was scientifically sound. 20

21 Tendency of a Noisy Signal x 21

22 Evolution of surface pressure before and after NNMI. (Williamson and Temperton, 1981) 22

23 Initialization of Richardson s Forecast Richardson s Forecast has been repeated on a computer. The atmospheric observations for 20 May, 1910, were recovered from original sources. ORIGINAL: INITIALIZED: dp s = +145 hpa/6 h dt dp s = 0.9 hpa/6 h dt Observations: The barometer was steady! 23

24 Richardson s Forecast Factory c François Schuiten 64,000 Computers: The first Massively Parallel Processor 24

25 The Finite Difference Scheme Let Q be governed by an equation dq dt = F (Q). The time interval under consideration is sliced into a finite number of discrete time steps {0, t, 2 t,..., n t,... }. The time derivitive is approximated by a finite difference: dq dt Q(t + t) Q(t t) 2 t. Thus, a problem in analysis becomes a problem in algebra. 25

26 Reversing History Differential calculus depends upon justifying the limiting process t 0. In approximating a differential equation, we reverse the procedure, and replace derivatives by ratios of increments. We thus... return to the manner in which they did things before the calculus was invented... (Richardson) 26

27 Stepping Forward The time derivative in dq dt = F (Q). is now approximated by a centered difference Q n+1 Q n 1 2 t = F n, Then Q n+1 = Q n t F n. This process of stepping forward is repeated a large number of times, until the desired forecast range is reached. We can discretize space in a similar way, but... 27

28 The Spectral Method The ECMWF Integrated Forecast System (IFS) uses a spectral representation of the meteorological fields. Each field is expanded in spherical harmonics, truncated at a fixed total wavenumber N: N n Q(λ i, φ j, t) = Q m n (t)yn m (λ i, φ j ) n=0 m= n The functions Y m n (λ, φ) are eigensolutions of the Laplacian: 2 Y m n = n(n + 1)Y m n. The coefficients Q m n (t) depend only on time. When the model equations are transformed to spectral space, they become a set of ordinary differential equations for the spectral coefficients Q m n. 28

29 ENIAC Forecast 29

30 The Meteorology Project Project estblished by John von Neumann in Objective of the project: To study the problem of predicting the weather using a digital electronic computer. A Proposal for Funding listed three possibilities : New methods of weather prediction Rational basis for planning observations Step towards influencing the weather! 30

31 The ENIAC 31

32 The ENIAC The ENIAC was the first multi-purpose programmable electronic digital computer. It had: 18,000 vacuum tubes 70,000 resistors 10,000 capacitors 6,000 switches Power: 140 kwatts 32

33 Charney, et al., Tellus, [ ] Absolute Vorticity = [ ] Relative Vorticity + [ ] Planetary Vorticity η = ζ + f. The atmosphere is treated as a single layer. The flow is assumed to be nondivergent. Absolute vorticity is conserved. d(ζ + f) dt = 0. This equation looks deceptively simple. But it is nonlinear: { } ψ 2 ψ t [ 2 ψ] + ψ 2 ψ + β ψ x y y x x = 0, 33

34 ENIAC Forecast for Jan 5,

35 NWP Operations The Joint Numerical Weather Prediction Unit was established on July 1, 1954: Air Weather Service of US Air Force The US Weather Bureau The Naval Weather Service. Operational numerical weather forecasting began in May, 1955, using a three-level quasi-geostrophic model. 35

36 Interlude 36

37 Temperature, Humidity and Climate Change 37

38 Data Assimilation 38

39 Data Assimilation NWP: An initial/boundary value problem Given an estimate of the present state of the atmosphere (initial conditions) appropriate surface and lateral boundary conditions the model forecasts the evolution of the atmosphere. Operational NWP centers produce initial conditions from a statistical combination of observations and short-range forecasts. This is called data assimilation. 39

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43 Optimal Interpolation The analysis problem is to find an optimum atmospheric state, x a, given A background field x b (on a regular grid) A set of (irregularly spaced) p observations y o The analysis is cast as background plus increment: x a = x b + W[y o H(x b )] The analysis and the background are vectors of length n. The weights are given by a matrix W of size (n p). 43

44 The Full Set of OI Equations The result of the (least squares) optimization is: x a = x b + W[y o H(x b )] W = BH T (R + HBH T ) 1 P a = (I WH)B All the covariance matrices are modelled using simplifying assumptions. Solution is a formidable computational task: The matrices are huge. Many shortcuts are needed. 44

45 Variational Assimilation Another approach to objective analysis is the variational assimilation technique. Problem: Find the analysis x that minimizes a cost function: J(x) = 1 2 { } (x x b ) T B 1 (x x b ) + [y o H(x)] T R 1 [y o H(x)] the distance between x and the background x b, plus the distance to the observations y o, : Variational assimilation has been shown to yield significant improvements in forecast accuracy. 45

46 The gradient of J with respect to x is J(x) = [B 1 + H T R 1 H](x x b ) H T R 1 {y o H(x b )} To find a minimum of J, we set J(x) = 0. The result is: x = x b + [ B 1 + H T R 1 H ] 1 H T R 1 {y o H(x b )} This is the (formal) solution of the 3-dimensional variational (3D-Var) analysis problem. The matrices are huge: perhaps

47 Minimization In practical 3D-Var, we do not invert a huge matrix. We find the minimum of J(x) by computing the cost function and using an optimization technique. The idea is to proceed downhill as fast as possible: Steepest Descent algorithm, Newton s method, Conjugate Gradient algorithm. 47

48 4D-Variational Assimilation Four-dimensional variational assimilation (4D-Var) is an extension of 3D-Var to allow for observations distributed within a time interval (t 0, t n ). The cost function includes terms for the distance to observations at the time of the observation. J[x(t 0 )] = 1 2 [x(t 0) x b (t 0 )] T B 1 0 [x(t 0) x b (t 0 )] N i=0 [ H(xi ) y o i ] T R 1 i The control variable is the initial state x (t 0 ). [ H(xi ) y o i ] 48

49 Schematic diagram of four dimensional variational assimilation. 49

50 Tangent Linear Model The solution at time t i+1 is computed from the solution at time t i by a (nonlinear) model: x i+1 = M i [x i ]. If we perturb the initial conditions, the solution is x i+1 + δx i+1 = M i [x i + δx i ] The linear tangent model is the (Jacobian) matrix: Then, to first order, [L i ] j,k = [M(x i)] j (x i ) k δx i+1 = L i δx i. 50

51 The Adjoint Model The transpose of the linear tangent model is called the adjoint model. The Gradient of the cost function is: J N [ ] = L T x 0 LT 1 LT i 1 0 i=0 H T i R 1 i d i Every iteration of the 4D-Var minimization requires the computation of the gradient: Compute the observation increments d i during a forward integration Multiply them by H T i R 1 i Integrate these weighted increments backward to the initial time using the adjoint model. 51

52 Atmospheric Normal Modes 52

53 Oscillations of the Atmosphere We treating the atmosphere as a thin single layer: du uv tan φ fv + g h dt a x = 0 dv dt + fu + u2 tan φ + g h a y = 0 ( dh u dt + h x + v y v tan φ ) = 0 a These are the shallow water quations on the sphere. We linearize about a motionless state with depth H: h(λ, φ, t) = Y (φ) exp[i(mλ σt)] After some algebra we get an equation for Y (φ): [( ) ] { [ ] d 1 µ 2 dy 1 m σ 2 + µ 2 dµ σ 2 µ 2 + dµ σ 2 µ 2 σ σ 2 µ 2 m2 1 µ 2 where µ = sin φ and ɛ = (2Ωa) 2 /gh. + ɛ } Y = 0. 53

54 Laplace Tidal Equation Again, the meridional structure is given by [( ) ] { [ ] d 1 µ 2 dy 1 m σ 2 + µ 2 dµ σ 2 µ 2 + dµ σ 2 µ 2 σ σ 2 µ 2 m2 1 µ 2 where µ = sin φ and ɛ = (2Ωa) 2 /gh. + ɛ } Y = 0. The normal modes are determined by the eigensolutions of this second order o.d.e., the Laplace Tidal Equation. Boundary conditions require Y to be regular at the poles. The Laplace Tidal Equation is not in Sturm-Lioville form. 54

55 Mathematical Difficulties The standard form of the Sturm-Liouville equation is ( d p(µ) dy ) + [q(µ) + λr(µ)]y = 0 dµ dµ where p(µ) is regular and has no zeros within the domain. For the Sturm-Liouville Equation: 1. The equation is self-adjoint and the eigenvalues λ are real. 2. The eigenfunctions for different λ are orthogonal. 3. The eigenfunctions form a complete set. 4. There is a denumerable infinity of non-negative eigenvalues with a single limit point at The zeros of the eigenfunctions behave according to the Sturmian oscillation theorems. 55

56 For the LTE, p(µ) = (1 µ 2 )/(σ 2 µ 2 ) blows up at the critical latutudes where µ = ±σ, and the equation is singular. Since the LTE cannot be written in standard Sturm-Liouville form, the five properties may not hold. It has been shown that the eigenvalues ɛ n of the LTE are real and the eigenfunctions form a complete, orthogonal set. Fourth and fifth properties do not hold. For σ < 1 there is a double infinity of eigenvalues, with limit points at both + and. The zeros of the eigenfunctions do not behave in a simple manner like for a regular Sturm-Liouville problem. 56

57 Eigenfrequencies σ of the LTE. 57

58 Atmospheric Predictability and Ensemble Forecasting 58

59 Progress in numerical weather prediction over the past fifty years has been quite dramatic. Forecast skill continues to increase... by one day per decade. However, there is a limit... 59

60 Chaos in Atmospheric Flow In a paper published in 1963, entitled Deterministic Nonperiodic Flow, Edward Lorenz showed that the solutions of the system Edward Lorenz (b. 1917) ẋ = σx + σy ẏ = xz + rx ż = +xy bz are highly sensitive to the initial conditions. 60

61 Identical Twin Experiment 61

62 Ensemble Forecasting In recognition of the chaotic nature of the atmosphere, focus has now shifted to predicting the probability of alternative weather events rather than a single outcome. The mechanism is the Ensemble Prediction System (EPS) and the world leader in this area is the European Centre for Medium-range Weather Forecasts (ECMWF). 62

63 Ensemble remains compact Ensemble spreads out 63

64 Singular Vectors The linear tangent model L i transforms a perturbation at time t i to a perturbation at time t i+1 : δx(t i+1 ) = L i δx(t i ) Perturbation growth is measured by the norm: δx(t i+1 ) 2 = L i δx(t i ) 2 = Lδx, Lδx = δx, L T Lδx. This depends on the eigenvalues of L T L, the singular values. The singular vector corresponding to the maximum singular value gives the component that grows fastest. 64

65 EPS: Ensemble Prediction System We calculate the 25 largest singular values, and the corresponding 25 singular vectors. Fifty perturbed initial states are constructed by adding and subtracting from the analysis. This gives us fifty-one initial states. Fifty-one forecasts are done, starting from these. 65

66 Ensemble of fifty-one 42-hour forecasts. Valid time: 0600 UTC, 26th December,

67 Discretizing the Sphere 67

68 Regular Latitude-Longitude Grid 68

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70 Conformal Stretched Grid 70

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72 Triangulated Icosahedral Grid 72

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78 Fibonacci Grid Inspired by Sun-flowers and Pineapples 78

79 The ultimate grid remains elusive. Ultimate depends on the application. 79

80 The End Typesetting Software: TEX, Textures, L A TEX, hyperref, texpower, Adobe Acrobat 4.05 Graphics Software: Adobe Illustrator L A TEX Slide Macro Packages: Wendy McKay, Ross Moore